Have to make some measurements in order to decide appropriate power supply and driver stage.

Signaali TL-201 educational laboratory PSU from early eighteens. Made in Finland.

I am a happy owner of a vintage power supply that is perfect for the job. It has a 0...20V output with a multiturn control pot and a voltage/current display. Checked its calibration with three different multimeters and it is still very precise. I did measure stopped motor current when applying a voltage between 4 and 10 volts in one volt steps. The slider doesn't move full travel with 3 volts.

V mA

4 210
5 260
6 310
7 360
8 410
9 460
10 510

I tried also to imitate an insane user by violently moving the knob back and forth. I saw only some 50...60 mA peaks added to each value. At 10 volts I managed to make the belt slip with that kind of violence. Anyway the plan is to apply a capasitive touch sensing of knob. So a lunatic user has to use a noncapasitive tool in order to cause that extra current.

Naturally the motor speed depends on voltage. I don't think that we would need scary fast faders in live sound. However the voltage must be high enough to ensure reliable movement for years on.

Conclusion

Motor operating voltage between 6...9 volts would be OK.
If you reserve 0.5A per fader in PSU, you are safe.
A standard H-bridge driver IC with 1A current capability is OK.